Density matrices are experimentally determined which describe H(n=2) atoms produced in electron-transfer collisions between 20-100-keV protons and helium. The density matrix contains the electron-transfer cross sections σ2s, σ2p0, and σ2p+/-1, as well as the real and imaginary parts of the s0p0 coherence. Experimentally, a monoenergetic proton beam traverses a helium gas cell producing hydrogen atoms H(n) via electron transfer. Within the gas cell an electric field is applied either axial or transverse to the proton beam. The Stokes parameters describing the intensity and linear polarization of Lyman-α radiation (122 nm) emitted by H(n=2) atoms are determined as a function of applied electric-field strength. The density-matrix elements are determined from a linear least-squares fit of the Stokes parameters to the set of five fitting functions which represent the contributions from individual density-matrix elements. The density-matrix results are self-consistent. Separate determinations using axial or transverse electric fields agree with each other. The general results indicate σ2s>σ2p0>σ2p+/-1 between 20 and 100 keV. The electric dipole moment <d>z has a value near zero at 20 keV rising to a maximum of about 1.3 a.u. near 40 keV and remaining nearly constant through 100 keV. The <L×A>z,s moment has a maximum of about 0.5 a.u. at 25 keV, passing through zero near 70 keV. These results compare favorably with available experimental results and are qualitatively predicted by present theoretical models. Comparison with previous H(n=3) results indicates that the Runge-Lenz vector <A>z is larger for n=3 than for n=2 and that <L×A>z,s has the same values for both n.